All the Star Trek canonical works (TV, film or, at a lower level, novels) only seem to give a very limited glimpse into the people in a starship design team (Leah Brahms at Utopia Planitia, Kurak for the Klingons). However, it stands to reason that starships are so complex that it would realistically require dozens, if not hundreds, of engineers to design any given class of starship.

Per Diplomatic Implausibility, we know Kurak led the design team of the Negh'Var-class.

@phantom42 - Agreed. There may be answers to this in lower canon novels or the canon & semi-canon reference books, cast or crew interviews, etc. Whoever's voting to close this is on a short ride to nowhere
– ValorumJun 3 '15 at 18:31

1 Answer
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Depending on your definition of "design team" the number is likely to be quite vague, taking into account the various teams responsible for the engines, computer core, propulsion systems that may be working on multiple projects.

For small ships (like the new runabout that Harry is designing in Voy: Non-Sequitur or the Delta Flyer) the teams appear to be relatively small, into the low tens of engineers with a single head of dept contributing the main design. For larger ships, the number is on a vast scale, with tens of thousands involved in the design, planning and construction phases across multiple planets.

To give you an idea, according to the Star Trek: TNG Technical Manual, the "Galaxy-Class Project" took two decades and required numerous systems to be imported from other Starfleet design teams working on theoretical system design

The history of the Galaxy Class Project, and of the USS Enterprise in
particular, is a story of technological innovation and teamwork
spanning more than twenty years. Research and fabrication centers
throughout the Federation, under the direct authority of Starfleet
Command's Advanced Starship Design Bureau (ASDB), combined their
efforts to plan and execute the newest and most complex vessel to join
Starfleet's inventory.

When the official start for the project was announced in July 2343,
much original theoretical work had already been accomplished,
particularly in the propulsion field. While the attempt to surpass the
primary warp field efficiency barrier with the Transwarp Development
Project in the early 2280s proved unsuccessful, the pioneering
achievements in warp power generation and field coil design eventually
led to the uprated Excelsior and Ambassador class starships.